Final answer:
To sketch the graph of the given equations, analyze the zeros and factors, plot the points, and consider the behavior around each zero. The equations can be factored to find the zeros and determine their multiplicity.
Step-by-step explanation:
To sketch the graph of the equation f(x) = 2(x+3)² (x−5), we need to analyze its factors and zeros. The equation can be factored as f(x) = 2(x^2 - 2x - 15). Setting each factor equal to zero, we find the zeros of the equation to be x = -3, x = -1, and x = 5. The graph will touch or cross the x-axis at these points, indicating multiplicity of 1 for each zero. We can plot these points on a graph and also consider the behavior around each zero to complete the sketch.
For the equation f(x) = x(x−1)⁴ (x+3)³, the zeros can be found by setting each factor equal to zero. We get the zeros as x = 0, x = 1, and x = -3. Again, the graph will touch or cross the x-axis at these points, indicating multiplicity of 1 for each zero. We can plot these points on a graph and analyze the behavior around each zero to complete the sketch.